Mathematical analysis for nonlinear partial differential equations with singular solutions

本研究的目的是解决非线性偏微分方程的解决方案，预计有依赖于时间的奇异性。奇点的候选者是调和映射中的缺陷、Ginzburg-Landau问题中的涡旋和自由边界。（1）在具有自由边界的肥皂膜振动中，建立了处理波动型自由边界问题的方法（2）发展了一种基于离散Mores流的体积约束数值方法（3）（4）构造了一个具有体积约束的抛物型方程的弱解，并证明了该解的Hoelder连续性。此外，我们还利用离散Morese流，发展了并行机极小化算法的求解器。这对于体积约束问题尤其有效。这对于弱连接的并行机也是非常好的，因为它使用了变分原理的直接方法。最后，我们要特别感谢这个项目的所有参与者。

The aim of this research was to solve nonlinear Partial Differential Equations whose solution is expected to have singularities depending on time. The candidates of singularities are defects in harmonic mapping, vortex in Ginzburg-Landau problem and free boundaries.We have solved the following problems;(1)On a Soap film vibration with free boundary, we have established the method to treat wave type free boundary problems(2)We developed a numerical method via the discrete Mores flow for volume constraint conditions(3)We constructed a weak solution to a hyperbolic equation with volume constraint(4)We constructed a weak solution to a parabolic equation with volume constraint and showing Hoelder continuity of thesolutionMoreover we have developed solvers for parallel machine with minimizing algorithm via the discrete Morese flows. This works very well especially for volume constraint problems. This is also very nice for a weak connected parallel machines, because it uses direct method of variational principle.Finally, we would like to express pur special thanks to all participants of this project.